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% 定义新的带灰色背景的说明环境 zremark
\newmdtheoremenv[
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  % 边框与背景一致，边框线会消失
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]{zremark}{说明}


\begin{document}
\title{15.6 注释}
\author{张志聪}
\maketitle

\section*{1}

\begin{zremark}
  三角不等式的推论：
  \begin{align*}
    |z - w| \geq  | \; |z| - |w| \; |
  \end{align*}
\end{zremark}

\textbf{证明：}

\begin{align*}
  |z| = |z - w + w| \leq |z - w| + |w| \\
  \implies                             \\
  |z - w| \geq |z| - |w|
\end{align*}

同理可得，
\begin{align*}
  |w - z| \geq |w| - |z|
\end{align*}

由于$|z - w| = |w - z|$，于是
\begin{align*}
  |z - w| \geq ||z| - |w||
\end{align*}

\section*{2}
\begin{zremark}
  任意实数$\epsilon > 0$，
  \begin{align*}
    |z - w| \leq \epsilon \\
    \implies              \\
    |w| - \epsilon \leq |z| \leq \epsilon + |w|
  \end{align*}
\end{zremark}

\textbf{证明：}

\begin{align*}
  |z| - |w| \leq |z - w| \leq \epsilon \\
  \implies                             \\
  |z| \leq \epsilon + |w|
\end{align*}

又
\begin{align*}
  |w| - |z| \leq |z - w| \leq \epsilon \\
  \implies                             \\
  |w| - \epsilon \leq |z|
\end{align*}

综上，
\begin{align*}
  |w| - \epsilon \leq |z| \leq \epsilon + |w|
\end{align*}

\end{document}